A173689 Numbers m such that the sum of square of factorial of decimal digits is square.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 122, 202, 212, 220, 221, 244, 424, 442, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 2222, 3333, 3444, 4344, 4434, 4443, 4444, 5555, 6666, 6677, 6767, 6776, 6888, 7667, 7676, 7766, 7777, 8688, 8868, 8886, 8888, 9999

Offset: 1

Author

Michel Lagneau, Nov 25 2010

Keywords

• nonn
• base

Comments

Let the decimal expansion of m = d(0)d(1)...d(p). Numbers such that Sum_{k=0..p} (d(k)!)^2 is square.

Examples

			a(16) = 244 is in the sequence because (2!)^2 + (4!)^2 + (4!)^2 = 1156 = 34^2.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1487

Crossrefs

Cf. A173687, A173688.

Programs

Maple

with(numtheory):for n from 0 to 10000 do:l:=length(n):n0:=n:s:=0:for m from
  1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s:=s+(u!)^2:od: q:=sqrt(s):if
  floor(q)= q then printf(`%d, `,n):else fi:od:

Mathematica

Select[Range[0,10000],IntegerQ[Sqrt[Total[(IntegerDigits[#]!)^2]]]&] (* Harvey P. Dale, Dec 19 2011 *)

Python

from itertools import count, islice, combinations_with_replacement
from math import factorial
from sympy.ntheory.primetest import is_square
from sympy.utilities.iterables import multiset_permutations
def A173689_gen(): # generator of terms
    yield 0
    for l in count(0):
        for i in range(1,10):
            fi = factorial(i)**2
            yield from sorted(int(str(i)+''.join(map(str,k))) for j in combinations_with_replacement(range(10), l) for k in multiset_permutations(j) if is_square(fi+sum(map(lambda n:factorial(n)**2,j))))
A173689_list = list(islice(A173689_gen(),50)) # Chai Wah Wu, Feb 23 2023

Extensions

Offset changed to 1 by Jinyuan Wang, Feb 26 2020